Abstract
The hypercube, though a popular and versatile architecture, has a major drawback in that its size must be a power of two. In order to alleviate this drawback, Katseff [1988] defined theincomplete hypercube, which allows a hypercube-like architecture to be defined for any number of nodes. In this paper we generalize this definition and introduce the namecomposite hypercube. The main result of our work shows that these incomplete architectures can be used effectively and without the size penalty. In particular, we show how to efficiently implement Fully Normal Algorithms on composite hypercubes. Development of these types of algorithms on composite hypercubes allows us to efficiently execute several algorithms concurrently on a complete hypercube. We also show that many host architectures, such as binary trees, arrays and butterflies, can be optimally embedded into composite hypercubes. These results imply that algorithms originally designed for any such host can be optimally mapped to composite hypercubes. Finally, we show that composite hypercubes exhibit many graph theoretic properties that are common with complete hypercubes. We also present results on efficient representations of composite hypercubes within a complete hypercube. These results are crucial in task allocation and job scheduling problems.
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Aleliunas, R., and Rosenberg, A. 1982. On embedding rectangular grids into square grids.IEEE Trans. Comps., C-31: 907–913.
Behzad, M., Chartrand, G., and Lesniak-Foster, L. 1979.Graphs and Diagraphs. Prindle, Weber, and Schmidt, Boston.
Bhatt, S.N., and Ipsen, I.C.F. 1985. How to embed trees in hypercubes. Res. rept. 443 (Dec), Dept. of Comp. Sci., Yale Univ., New Haven, Conn.
Bhatt, S., Chung, F., Leighton, F.T., and Rosenberg, A. 1986. Optimal simulations of tree machines. InProc., 27th Symp. on the Foundations of Comp. Sci., pp. 274–282.
Boals, A., Gupta, A., and Sherwani, N. 1992. Incomplete hypercubes: Embeddings and algorithms. Tech. rept. TR/92-22, Dept. of Comp. Sci., W. Mich. Univ., Kalamazoo, Mich.
Bondy, J.A., and Murthy, U.S.R. 1976.Graph Theory with Applications. MacMillan, New York.
Chan, M.Y. 1988. Dilation-2 embeddings of grids into hypercubes. Tech. rept. UTDCS 1-88, Dept. of Comp. Sci., Univ. of Tex. at Dallas.
Chan, M.Y., and Lee, S.-J. 1993. Fault-tolerant permutation routing in hypercubes.J. Parallel and Distr. Comp. (Apr.): 227–281.
Chen, H.L., and Tzeng, N.-F. 1989. Enhanced incomplete hypercubes. InProc., Internat. Conf. on Parallel Processing, vol. 1, pp. 270–277.
Chen, M., and Shin, K.G. 1987. Processor allocation in a N-CUBE multiprocessor using Gray codes.IEEE Trans. Comps., C-36, 12: 1396–1407.
Chen, M., and Shin, K.G. 1988. Message routing in an injured hypercube. InProc., 3rd Conf. on Hypercube Concurrent Comps. and Applications, pp. 312–317.
Das, S.K., Deo, N., and Prasad, S. 1990. Parallel graph algorithms for hypercube computers.Parallel Computing, 13: 143–158.
Fishburn, J.P., and Finkel, R.A. 1982. Quotient networks.IEEE Trans. Comps., C-31, 4: 288–295.
Gordon, J.M., and Stout, Q.F. 1988. Hypercube message routing in the presence of faults. InProc., 3rd Conf. on Hypercube Concurrent Comps. and Applications, pp. 318–327.
Graham, N., Harary, F., Livingston, M., and Stout, Q.F. 1993. Subcube fault-tolerance in hypercubes.Information and Computation (Feb.): 218–314.
Greenberg, D.S., Heath, L.S., and Rosenberg, A.L. 1990. Optimal embeddings of butterfly-like graphs in the hypercubes.Math Systems Theory, pp. 61–77.
Gupta, A.K. 1989. On the relationship between parallel computation and graph embeddings. Ph.D. thesis, Purdue Univ., Lafayette, Ind.
Harary, F., Hayes, J.P., and Wu, H.-J. 1988. A survey of the theory of hypercube graphs.Computational Math. and Applications, 15, 4: 277–289.
Hastad, J., Leighton, F.T., and Newman, M. 1987. Reconfiguring a hypercube in the presence of faults. InProc., 19th Annual ACM Symp. on the Theory of Computing, pp. 274–284.
Hastad, J., Leighton, T., and Newman, M. 1989. Fast computation using faulty hypercubes. InProc., 21st Annual Symp. on the Theory of Computing, pp. 251–263.
Havel, I., and Móravek, J. 1972. B-valuations of graphs.Czech Math. J., 22: 338–351.
Ho, C.T., and Johnsson, S.L. 1990. Embedding meshes in Boolean cubes by graph decomposition.J. Parallel and Distr. Comp., 8: 325–339.
Hong, J.W., Mehlhorn, K., and Rosenberg, A. 1983. Cost trade-offs in graph embeddings, with applications.JACM: 709–728.
Kandlur, D.D., and Shin, K.G. 1988. Hypercube management in the presence of node failures. InProc., 3rd Conf. on Hypercube Concurrent Comps. and Applications, pp. 328–336.
Katseff, H.P. 1988. Incomplete hypercubes.IEEE Trans. Comps., 37, 5: 604–608.
Kosaraju, S.R., and Atallah, M. 1988. Optimal simulations between mesh connected array of processors.JACM, 35, 3: 635–650.
Lee, T.C., and Hayes, J.P. 1988. Routing and broadcasting in faulty hypercube computers. InProc., 3rd Conf. on Hypercube Concurrent Comps. and Applications, pp. 346–354.
Leiserson, C.E. 1980. Area-efficient graph layouts (for VLSI). InProc., 22nd Annual IEEE Symp. on the Foundations of Comp. Sci., pp. 270–281.
Prabhalla, V., and Sherwani, N. 1990. Parallel single row routing on compact hypercubes. Tech. rept. TR/90-06, Dept. of Comp. Sci., W. Mich. Univ., Kalamazoo, Mich.
Roy, A., Deogun, J.S., and Sherwani, N.A. 1989. A parallel algorithm for single row routing problems.J. Circuits, Systems, and Comps. (to appear).
Saad, Y., and Schultz, M.H. 1988. Topological properties of hypercubes.IEEE Trans. Comps., 37, 7: 867–872.
Tien, J.-Y., and Yang, W.-P. 1991. Hierarchical spanning trees and distributing on incomplete hypercubes.Parallel Comp., 17: 1343–1360.
Tzeng, N.-E 1990. Structural properties of incomplete hypercubes. InProc., 10th Internat. Conf. on Distr. Computing Systems, pp. 262–269.
Tzeng, N.-E, Chen, H.L., and Chuang, P.J. 1990. Embeddings in incomplete hypercubes. InProc., Internat. Conf. on Parallel Processing, pp. III-335–III-339.
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This research was supported in part by the National Science Foundation under grant USE-90-52346. A preliminary version of this work appeared in the5th International Parallel Processing Symnposium, May 1991.
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Boals, A.J., Gupta, A.K. & Sherwani, N.A. Incomplete hypercubes: Algorithms and embeddings. J Supercomput 8, 263–294 (1994). https://doi.org/10.1007/BF01204731
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DOI: https://doi.org/10.1007/BF01204731